Discovery of enhanced lattice dynamics in a single-layered hybrid perovskite

Layered hybrid perovskites exhibit emergent physical properties and exceptional functional performances, but the coexistence of lattice order and structural disorder severely hinders our understanding of these materials. One unsolved problem regards how the lattice dynamics are affected by the dimensional engineering of the inorganic frameworks and their interaction with the molecular moieties. Here, we address this question by using a combination of spontaneous Raman scattering, terahertz spectroscopy, and molecular dynamics simulations. This approach reveals the structural dynamics in and out of equilibrium and provides unexpected observables that differentiate single- and double-layered perovskites. While no distinct vibrational coherence is observed in double-layered perovskites, an off-resonant terahertz pulse can drive a long-lived coherent phonon mode in the single-layered system. This difference highlights the dramatic change in the lattice environment as the dimension is reduced, and the findings pave the way for ultrafast structural engineering and high-speed optical modulators based on layered perovskites.

butylammonium bromide (98%, SIGMA-ALDRICH) at room temperature. As shown in Figure. S6, the TKE signal from n-butylammonium bromide also displays a strong bipolar response, similar to what is observed in the n=1 2DHP. Consequently, it appears plausible that, alongside the attribution to the nonlinear propagation effect, the observed bipolar response in the n=1 2DHP could be influenced by the presence of the n-butylammonium (BA) spacers. However, this bipolar response is not observed in the n=2 2DHP, which has an additional lead bromide octahedral layer and more crucially, the methylammonium organic cations. Therefore, it is likely that this more complicated structure of the n=2 system leads to the suppression of the bipolar response.

Supplementary Note 3: Sum-frequency excitation of the Raman mode
In this section, we provide a comparison of the two nonlinear excitation pathways to drive the Raman mode. For a two-photon Raman scattering process, we assume a harmonic lattice potential V (Q R ) = 1 2 Ω 2 R Q 2 R , and the corresponding Lagrange equation with a classical Raman type harmonic-oscillator can be derived as (47,48) ( Here, Q R represents the normal Raman mode coordinate, Γ R is a phenomenological damping term, Ω R is the eigenfrequency of the Raman mode, E(t) represents the driving electric field, and χ is the linear dielectric susceptibility. The excitation of the Raman mode is mediated by the Raman tensor ∂χ ∂Q R . Since the driving term on the right-hand side of the equation scales as the square of the pump electric field, which can couple to the Raman mode through either difference-(i.e., Ω 1 − Ω 2 = Ω R ) or sum-frequency (i.e., Ω 1 + Ω 2 = Ω R ) components of light, this equation of motion describes impulsive stimulated Raman scattering as well as sum-frequency excitation observed here. It is worth noting that the dielectric responses of hybrid perovskites feature large jumps in the THz range as the frequency decreases across several broad transverse optical phonon resonances. Since for both Raman excitation processes, the pump electric field interacts with virtual electronic dipole transitions, THz off-resonance excitation gives rise to colossal nonlinear polarizability response compared to that in the optical range. This can be also viewed in the time domain as a cloud of electrons bound to a nucleus displaces more strongly in response to a slowly varying electric field as compared to the ultrafast optical pump pulses. The non-impulsive nature of the THz sum-frequency excitation mechanism may also explain why the THz pump pulse selectively drives the octahedral twist mode with the largest polarizability. The quasi-DC electric field continuously builds up the phonon amplitude, whereas in the optical impulsive stimulated Raman scattering process, all Raman phonons within the pump bandwidth are coherently excited (34).
In contrast, the ionic Raman scattering requires an anharmonic lattice potential and its simplest form can be described as where c is the anharmonic coupling coefficient. The corresponding equations of motion are (43) where in the first equation Z IR is the effective charge of the infrared-active phonon mode. In this case, the Raman mode is activated by anharmonic coupling to the directly driven infrared-active mode. For this process to be efficient, there should exist an infrared-active phonon mode with its eigenfrequency that matches the sum-frequency excitation condition (Ω IR = 1 2 Ω R , i.e., ∼ 0.9 THz), which is ruled out by the timedomain THz spectroscopy measurement. Therefore, we confirm that the driven Raman mode excited through large polarizability rather than anharmonicity.

Supplementary Note 4: Decay process of the driven Raman mode
In this section, we provide additional details on the fitting and interpretation of the temperature-dependent decay rates of the driven Raman mode in the n=1 sample. As shown in Figure 2D, a notable reduction in the decay rate is observed with decreasing temperature. The raw data, along with the fits, are presented in Figure. S7.
In general, the coherent phonon decay process involves a combination of anhar-monic decay, which occurs via anharmonic coupling to acoustic phonons, and pure dephasing. To obtain a more comprehensive understanding of the temperature-dependent decay rates, we fit the data to the following equation: where the first term represents the temperature-dependent anharmonic decay rate (79).
Here, Γ 0 is the effective anharmonic decay constant, Ω R is the frequency of the Raman mode, and k B is the Boltzmann constant. The second term corresponds to the temperature-independent phonon decay rate, such as via phonon-defect scattering processes. The fits agree well with the experimental data, with Γ 0 and Γ 1 determined to be 0.026 ps −1 and 0.059 ps −1 , respectively. These results suggest that the decay of the Raman coherence is dominated by the anharmonic decay channel at high temperatures.
Previous studies have shown that 2DHPs feature optically-inactive acoustic branches as well as zone-folded longitudinal acoustic phonons inherent to the alternating layers (22), which may also contribute to the efficient decay of the Raman coherence.
Note that our findings appear to contradict a previous study that employed transient absorption spectroscopy with impulsive stimulated Raman scattering (34). This earlier investigation did not observe any long-lived phonon coherence in (BA) 2 PbBr 4 and concluded that a temperature-independent decay rate existed based on this observa-tion. However, our study, which used THz sum-frequency excitation, reveals long-lived Raman coherence was not observed in the previous study using impulsive stimulated Raman scattering and highlights the strength of using intense THz pulses to coherently drive low-energy collective modes in hybrid perovskites. As for the MD simulation, Ref.
(34) only considered the lowest frequency optical phonon mode, which may couple to the organic ligands more efficiently than the octahedral twist mode discussed here.

A. Finite temperature calculations
We used a Nosé-Hoover thermostat reference (adapted from LAMMPS) to sample the thermodynamics of the system, where the thermal damping time and targeted temperature were set at 1 ps and 77 K, respectively. (80) To ensure a high-frequency resolution, we calculated the auto-correlation function using 600 steps, yielding a frequency resolution of approximately 0.017 THz, which is much narrower than any Raman responses of interest. Figure. S8 shows the time evolution of the non-zero tensor elements of the dielectric susceptibility in thermal equilibrium along with the temperature fluctuations (See inset plots). The spontaneous Raman scattering intensity was calculated based on the isotropic average condition (68), where I // and I ⊥ denote the Raman scattering intensity polarized parallel and perpendicular to the incident light polarization; ω in , ω p are the frequencies of the incident and scattered light; and Q p represents the normal mode coordinate; a p is the isotropic polarizability, defined as and γ p is the anisotropic polarizability, defined as The Raman tensor ∂χ ij ∂Qp was calculated by computing the time-domain auto-correlation We also conducted further tests to evaluate the influence of thermostat parameters, specifically adjusting the damping time to 0.1 ps. As depicted in Fig. S9, the recalculated Raman spectra for both 2DHPs demonstrate qualitative and quantitative alignment with those shown in Fig. 4A of the main text. This consistency in results across different damping parameters serves to underscore the robustness of our simulations, thereby affirming that our findings are indeed invariant to the specific choice of damping parameter used.

B. Real-space analysis
To identify the lattice displacements corresponding to the Raman peaks, we filtered the trajectories of the system ⃗ X(t) with each peak frequency Ω R and a window ∆ω to get the real-space trajectory with the mode frequency equals to Ω R : where Θ is the Heaviside step function, and FT denotes the Fourier transform. We THz modes correspond to the octahedral bending and twisting motions.

C. Simulation of organic cation ordering
To evaluate the impact of molecular moieties on the dynamical disorder, we compute the auto-correlation function of the ordering of each organic ligand chain for both the n=1 and n=2 systems. Specifically, we define the auto-correlation function as follows: Here, ⃗ d represents a vector indicating the direction of each organic cation from head to tail. Figure S10  In this section, we provide an estimate of the Kerr nonlinear coefficient and the refractive modulation depth of the n=1 2DHP under the intense THz fields. When irradiating the material with a peak THz electric field strength of 610 kV/cm at room temperature, we observe a 1.2% deviation from the balanced signals at the arrival of the THz peak. In the balanced detection scheme using a half-wave plate (81), the differential signal is where I 1 and I 2 are intensities of the two orthogonally polarized beams measured by a